Calculus Examples

$\sin (2 x)$

Step 1

Let

$u = 2 x$ . Then

$d u = 2 d x$ , so

$\frac{1}{2} d u = d x$ . Rewrite using

$u$ and

$d$

$u$ .

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Step 1.1

Let

$u = 2 x$ . Find

$\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate

$2 x$ .

$\frac{d}{d x} [2 x]$

Step 1.1.2

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x$ with respect to

$x$ is

$2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$2 \cdot 1$

Step 1.1.4

Multiply

$2$ by

$1$ .

$2$

$2$

Step 1.2

Rewrite the problem using

$u$ and

$d u$ .

$\int \sin (u) \frac{1}{2} d u$

$\int \sin (u) \frac{1}{2} d u$

Step 2

Combine

$\sin (u)$ and

$\frac{1}{2}$ .

$\int \frac{\sin (u)}{2} d u$

Step 3

Since

$\frac{1}{2}$ is constant with respect to

$u$ , move

$\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \sin (u) d u$

Step 4

The integral of

$\sin (u)$ with respect to

$u$ is

$- \cos (u)$ .

$\frac{1}{2} (- \cos (u) + C)$

Step 5

Simplify.

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Step 5.1

Simplify.

$\frac{1}{2} (- \cos (u)) + C$

Step 5.2

Combine

$\frac{1}{2}$ and

$\cos (u)$ .

$- \frac{\cos (u)}{2} + C$

$- \frac{\cos (u)}{2} + C$

Step 6

Replace all occurrences of

$u$ with

$2 x$ .

$- \frac{\cos (2 x)}{2} + C$

Step 7

Reorder terms.

$- \frac{1}{2} \cos (2 x) + C$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$